Properties

Label 2-1440-20.19-c2-0-46
Degree $2$
Conductor $1440$
Sign $-0.453 + 0.891i$
Analytic cond. $39.2371$
Root an. cond. $6.26395$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 + 4.75i)5-s + 0.206·7-s − 15.0i·11-s + 11.6i·13-s − 18.1i·17-s − 19.3i·19-s − 27.2·23-s + (−20.2 + 14.7i)25-s − 44.4·29-s + 20.3i·31-s + (0.319 + 0.982i)35-s + 18.1i·37-s + 32.3·41-s + 4.06·43-s + 5.37·47-s + ⋯
L(s)  = 1  + (0.309 + 0.950i)5-s + 0.0295·7-s − 1.36i·11-s + 0.899i·13-s − 1.07i·17-s − 1.02i·19-s − 1.18·23-s + (−0.808 + 0.588i)25-s − 1.53·29-s + 0.657i·31-s + (0.00913 + 0.0280i)35-s + 0.489i·37-s + 0.787·41-s + 0.0944·43-s + 0.114·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.453 + 0.891i$
Analytic conductor: \(39.2371\)
Root analytic conductor: \(6.26395\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1),\ -0.453 + 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8004350312\)
\(L(\frac12)\) \(\approx\) \(0.8004350312\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.54 - 4.75i)T \)
good7 \( 1 - 0.206T + 49T^{2} \)
11 \( 1 + 15.0iT - 121T^{2} \)
13 \( 1 - 11.6iT - 169T^{2} \)
17 \( 1 + 18.1iT - 289T^{2} \)
19 \( 1 + 19.3iT - 361T^{2} \)
23 \( 1 + 27.2T + 529T^{2} \)
29 \( 1 + 44.4T + 841T^{2} \)
31 \( 1 - 20.3iT - 961T^{2} \)
37 \( 1 - 18.1iT - 1.36e3T^{2} \)
41 \( 1 - 32.3T + 1.68e3T^{2} \)
43 \( 1 - 4.06T + 1.84e3T^{2} \)
47 \( 1 - 5.37T + 2.20e3T^{2} \)
53 \( 1 + 79.1iT - 2.80e3T^{2} \)
59 \( 1 + 83.3iT - 3.48e3T^{2} \)
61 \( 1 + 36.7T + 3.72e3T^{2} \)
67 \( 1 - 4.51T + 4.48e3T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 15.5iT - 6.24e3T^{2} \)
83 \( 1 + 50.9T + 6.88e3T^{2} \)
89 \( 1 + 10.8T + 7.92e3T^{2} \)
97 \( 1 - 12.1iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.242819048113952835320901114250, −8.274342923108272886956024324322, −7.35548974971844648455441896580, −6.61472967291891458656711664213, −5.92164307450560261954588716026, −4.95278147364991328555022253694, −3.73364826338102481153247477045, −2.93277066136664978209034109941, −1.88305036305111290972540833448, −0.21218665288960495769378857195, 1.40677080180912845535223061696, 2.24114717593397093011791419016, 3.84148527231534127864915732242, 4.45780959602155916111940336104, 5.64065849248779684451988476093, 6.00333788851436005818152240984, 7.45541630481671915353801088422, 7.928008537323925449646710441746, 8.809639165511330440943479428912, 9.735637337128141397331994458813

Graph of the $Z$-function along the critical line